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Properties

Label 1-183-183.182-r1-0-0
Degree $1$
Conductor $183$
Sign $1$
Analytic cond. $19.6660$
Root an. cond. $19.6660$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 17-s + 19-s − 20-s + 22-s + 23-s + 25-s + 26-s − 28-s + 29-s − 31-s + 32-s + 34-s + 35-s − 37-s + 38-s − 40-s − 41-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 17-s + 19-s − 20-s + 22-s + 23-s + 25-s + 26-s − 28-s + 29-s − 31-s + 32-s + 34-s + 35-s − 37-s + 38-s − 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $1$
Analytic conductor: \(19.6660\)
Root analytic conductor: \(19.6660\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{183} (182, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 183,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.195656533\)
\(L(\frac12)\) \(\approx\) \(3.195656533\)
\(L(1)\) \(\approx\) \(1.857865691\)
\(L(1)\) \(\approx\) \(1.857865691\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
61 \( 1 \)
good2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.030079139609153032826145091820, −25.77718706830122597355147197810, −25.053423918135471749965502732589, −23.91972303826479248927248247671, −22.973611132598616197671802772020, −22.60582542477840033559738202447, −21.387320540039787672584244049552, −20.24404977648919722563945703063, −19.55680173499314007357158387406, −18.64191173710907383996537849279, −16.754153814892083935896283941462, −16.12067827710113812627045770249, −15.238364324195480430921721067951, −14.19376663011857738138526428338, −13.126308416446661210608302989546, −12.12191384536772732308634901298, −11.430975381800504799070590330365, −10.164559626679759920692313524708, −8.678961664768601566938244711441, −7.26817830294184876752803053047, −6.46836157260039145452729188588, −5.15992567650912984316431048855, −3.675484790853870935437939468864, −3.27142393546728453666476919051, −1.13287869528821040848638909564, 1.13287869528821040848638909564, 3.27142393546728453666476919051, 3.675484790853870935437939468864, 5.15992567650912984316431048855, 6.46836157260039145452729188588, 7.26817830294184876752803053047, 8.678961664768601566938244711441, 10.164559626679759920692313524708, 11.430975381800504799070590330365, 12.12191384536772732308634901298, 13.126308416446661210608302989546, 14.19376663011857738138526428338, 15.238364324195480430921721067951, 16.12067827710113812627045770249, 16.754153814892083935896283941462, 18.64191173710907383996537849279, 19.55680173499314007357158387406, 20.24404977648919722563945703063, 21.387320540039787672584244049552, 22.60582542477840033559738202447, 22.973611132598616197671802772020, 23.91972303826479248927248247671, 25.053423918135471749965502732589, 25.77718706830122597355147197810, 27.030079139609153032826145091820

Graph of the $Z$-function along the critical line